\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros


\def\lf{\ \hfil\break}       % Neue Zeile ohne Einr"ucken, 'linefeed'
\def\cl{\centerline}
\def\LF{\medskip\noindent}   % Neue Zeile mit breiterem Zwischenraum


\nopagenumbers

\vglue 10pt

\Title Ellipse.

\medskip


\LF
See also:\hfil\break Parabola, Hyperbola, Conic Sections and their ATOs.

\LF
The usual parametric representation for an Ellipse with semi-axes $aa$
and $bb$ is:

\lf
$x(t) = aa \cos(t)  \lf
 y(t) = bb \sin(t) ,\hskip1cm  0\le t \le 2\pi$.

\lf
(Here, $aa$ and $bb$ are assumed positive. The larger is called the 
semi-major axis length,  the smaller the semi-minor axis length.)
The corresponding implicit equation is: \hfil\break
\cl{$ ({x^2\over aa})^2 + ({y\over bb})^2 = 1$.}

\noindent
The following geometric definition of an Ellipse can be used to
create elliptical flower beds: 

{\narrower \noindent
An Ellipse is the set of points for which the {\bf sum of the distances}
from two focal points is a constant $L$ equal to twice the semi-major 
axis length. \par}

\noindent
A gardener connects the two focal points by a cord of length $L$, pulls the cord 
tight with a stick, then draws the boundary of the flower bed
with the stick. 

\noindent
Another version of this definition is: 

{\narrower \smallskip\noindent 
An Ellipse is the set of
points which have {\bf equal distance} from a circle of radius $L$
and a (focal) point inside the circle.\smallskip} 

\noindent
Both these definitions are illustrated in the program. 

\noindent
The normal to an ellipse at any point bisects the angle made by the
two lines joining that point to the foci.
This says that rays coming out of one focal point are
reflected off the ellipse towards the other focal point. Therefore one can
build elliptically shaped ``whispering galleries'', where a word spoken 
softly at one focal point can be heard only close to the other
focal point.

Here is a simple proof that an ellipse lies on one side of any tangent or, 
more precisely, that for every point on the tangent other than the point of tangency,
the sum of the distances to the two focal points $F_1, F_2$
is greater than the length $L$ of the major axis. (In the display: $F=F_2$.)
Pick any point $Q$ on the tangent, join it to the two focal points
and reflect the segment $QF$ in the tangent, giving another segment $QS$.
Now $F_1QS$ is a radial straight segment only if $Q$ is the point of 
tangency---otherwise
$F_1QS$ is by the triangle inequality longer than the radius $F_1S$
(of length $L$) of the circle around $F_1$.


\lf
The evolute of an ellipse (i.e., the curve enveloped by the normals of the
ellipse---see Action Menu: Draw osculating circles with normals) is a
generalized Astroid.  If one chooses in the Settings Menu  $aa/bb =: x$ 
where $ x\sim 1.4656 $ is the solution of $x^2(x-1) = 1,$ then the evolute is the standard Astroid.

\lf
An Ellipse can also be obtained by a rolling construction: Inside a circle of
radius $aa$ another circle of radius r  = hh = 0.5*aa rolls and traces
the Ellipse with a stick of radius R = ii*r. The parametric equation resulting
from this construction is:

\lf
$ x(t) = (R + r)*\cos(t) \lf
  y(t) = (R - r)*\sin(t) $

\lf
This is related to the visualization of the complex map $z\to z+1/z$ in Polar
Coordinates, the image of the circle of radius $R$ is such an ellipse with
$r=1/R$.
\lf
Such rolling constructions are reached with the Menu entry `Circle' and then
the Action Menu `Draw Generalized Cycloids'. Recall that negative values of the
rolling radius $hh$ gives curves on the outside, positive radii ($hh < aa$)
on the inside of the fixed circle. \lf
Other rolling curves are:  \lf
Cycloid, Astroid, Cardioid, Limacon, Nephroid.

\noindent
H.K.



\bye
